伴随幂零理想与时排列

Let g be a complex simple Lie algebra. Let b be a fixed Borel subalgebra of g. The study of abelian ideals of b goes back to Kostant. For example, he related it to the representation theory of semisimple Lie groups, the cohomology of Kac-Moody algebras, and the Ramanujan numbers etc. This motivated people to study more general objects, i.e, ad-nilpotent ideals of b..Although phrased in a slightly different language, Jian-Yi Shi had essentially given a natural bijection between the set of all the ad-nilpotent ideals of b and the set of all the dominant regions of the now-called Shi arrangement. This is called Shi bijection. In a preprint, we extend this result by establishing a bijection between the set of all the ad-nilpotent ideals of a parabolic subalgebra of g and the set of all the dominant regions of certain deleted Shi arrangement Shi(I)..When I is empty, Shi(I) recovers the Shi arrangement; when I equals to the set of all simple roots, Shi(I) recovers the Coxeter arrangement. Note that both Coxeter and Shi arrangements are important examples for hyperplane arrangements..In this applied project, our first target is to investigate Shi(I) more carefully. For example, we want to study its characteristic polynomial. We want to ask whether it is free..On the other hand, we want to see what regions shall correspond to abelian ideals of b under the Shi bijection. This would give us a vivid way to present the abelian ideals of b. .More generally, we want to see what regions shall correspond to abelian ideals of a parabolic subalgebra under the extended Shi bijection. We want to see that do they form a beautiful structure.

设g是一个复的单李代数，b是g的一个取定的 Borel子代数。 对b的Abel理想的研究可以追溯到Kostant。他把这一话题与半单李群的表示理论、Kac-Moody代数的上同调以及拉马努金数等联系了起来。Kostant的工作促使人们去考虑更广泛的研究对象：b的伴随幂零理想。.尽管所用的语言稍有差异，时俭益实际上已经在b的伴随幂零理想集和现在称为时排列的支配区域集之间建立了一个自然的双射。这一结果称为时双射。在一份预印本中，我们建立了g的抛物子代数的伴随幂零理想集和Shi(I)的支配区域集之间的一个自然的双射。这里I是单根集的一个子集，Shi(I)是由时排列Shi去掉适当的超平面得来。当I为空集时，Shi(I)即为时排列；当I为单根集时，Shi(I)即为Coxeter排列。. 在本项目中，我们拟深入研究Shi(I)的组合性质，以及它的哪些支配区域对应于抛物子代数的Abel理想。

我们从李代数的背景出发对时排列展开了深入的研究，提出了拟反链的概念，并由此得到了一般的时排列的特征多项式的交错和公式。拟反链这个概念是分拆在李代数意义下的推广，由此出发能推广经典的第二类Stirling数等。我们还针对拟反链提出了折叠式鼓图这个概念，它是拟反链这一代数概念的组合模型。借助该模型，我们对已有的几类从非交叉分拆到非叠合分拆的双射给予了统一的解释，并推广了一些已知的结果。我们在这方面的工作发表在Journal of Combinatorial Theory-Series A和Advances in Applied Mathematics上。..在狄拉克上同调方面，我们整理了自己的博士学位论文并发表在Transformation Groups上。在这个工作的基础上，我们部分解决了Barbasch和Pandzic提出的一个猜测，相关论文发表在Communications in Algebra上。..另外，我们与美国加州大学San Francisco分校的研究人员合作完成了一篇应用数学方面的论文，投稿至牛津大学出版社出版的学术期刊Mathematical Medicine and Biology。..最后，本项目部分资助了一名博士研究生和一名硕士研究生的培养。